To find the area and perimeter of the figure determined by the points $(-3, -1)$, $(-2, 3)$, $(6, 3)$, and $(5, -1)$, let's break the problem down step by step.

1. Find the Area

These points represent a quadrilateral, and we can calculate the area using the Shoelace Theorem (or the formula for the area of a polygon given its vertices).

The vertices of the quadrilateral are:

• $(-3, -1)$
• $(-2, 3)$
• $(6, 3)$
• $(5, -1)$

Using the Shoelace Theorem: $\text{Area} = \frac{1}{2} \left| x_1y_2 + x_2y_3 + x_3y_4 + x_4y_1 - (y_1x_2 + y_2x_3 + y_3x_4 + y_4x_1) \right|$ Substitute the values: $\text{Area} = \frac{1}{2} \left| (-3)(3) + (-2)(3) + 6(-1) + 5(-1) - \left[(-1)(-2) + 3(6) + 3(5) + (-1)(-3)\right] \right|$

2. Find the Perimeter

The perimeter is the sum of the lengths of the sides of the quadrilateral. We can use the distance formula to find the length of each side:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

The sides are between the points:

• $(-3, -1)$ and $(-2, 3)$
• $(-2, 3)$ and $(6, 3)$
• $(6, 3)$ and $(5, -1)$
• $(5, -1)$ and $(-3, -1)$

Let me calculate both the area and the perimeter for you.The area of the figure is $32$ square units, and the perimeter is approximately $24.25$ units.

Would you like further clarification or details on any part of the solution?

Here are 5 related questions for further practice:

1. How would the area change if one of the vertices was moved?
2. What is the formula for the Shoelace Theorem, and how does it apply to polygons with more sides?
3. How does the perimeter change if the figure becomes a regular polygon?
4. Can you calculate the diagonal lengths for this quadrilateral?
5. How does the area of this figure compare to a rectangle with the same side lengths?

Tip: When calculating the perimeter of a polygon, always apply the distance formula carefully between consecutive vertices!